Removing Trapped Rubidium through Optical Pumping at Large Magnetic Fields

by

Conrad Newfield

B.S., Engineering Physics, University of Colorado Boulder

B.S., Applied Mathematics, University of Colorado Boulder

Defense Date: 04/13/2020

Research Advisor: Jun Ye (Physics)

Honors Council Representative: Paul Beale (Physics)

Committee Member: Michael Litos (Physics)

Committee Member: Brian Zaharatos (Applied Mathematics)

A thesis submitted to the Faculty of the

University of Colorado in partial fulfillment

of the requirements for the award of

Departmental Honors in the

Department of Physics

2020 This thesis entitled: Removing Trapped Rubidium through Optical Pumping at Large Magnetic Fields written by Conrad Newfield has been approved for the Department of Physics

Prof. Jun Ye

Prof. Michael Litos

Prof. Paul Beale

Prof. Brian Zaharatos

Date

The final copy of this thesis has been examined by the signatories, and we find that both the content and the form meet acceptable presentation standards of scholarly work in the above mentioned discipline. iii

Newfield, Conrad (B.S., Engineering Physics)

Removing Trapped Rubidium through Optical Pumping at Large Magnetic Fields

Thesis directed by Prof. Jun Ye

Ultracold dipolar molecules like 40K87Rb are ideal candidates to study phenomena in quantum chemistry and many-body physics. The additional rotational and vibrational degrees of freedom of molecules make them interesting species to study but also create huge technological challenges. Due to the experimental complexity and large number of cooling steps required for molecule production at nanokelvin temperatures, the final molecule count for our experiment fluctuates and composes only a small fraction of the initial atom count. Blasting away excess atoms during initial molecule production is key to augmenting number consistency. KRb will be destroyed if unconverted K and

Rb atoms are not removed quickly enough. To improve molecule number and stability, a method of blasting Rb at high magnetic fields was tested by optically pumping rather than using radio- frequency adiabatic rapid passages (ARPs). Ground-state KRb lifetime was found to be the same for both methods. This thesis describes this attempt at optical pumping, its theoretical basis, and results. Dedication

To friends, at CU and afar, family, and clean basalt geometry. v

Acknowledgements

My journey to end to up working in a physics lab started early in life. Several incredible teachers guided me and encouraged my interest in math and physics from a young age. Specifically

I would like to thank Laura Cummings, auntie Betsy Hatter, and Tiffany Coke.

I certainly wouldn’t have ended up in Boulder without my passion for climbing, so I must thank the Hawai‘i climbing community.

I would like to thank Michael Litos for guiding me throughout college and helping me get a position in one of Jun Ye’s labs. I would of course like to thank Jun because none of this work would have been possible without his belief in me. Jun’s work ethic is otherworldly and I feel incredibly lucky to have been able to work under him. I can’t wait to see all the groundbreaking research that comes out of his labs in the next decades.

I would also like to thank the graduate students and postdocs who served as my mentors in lab: Kyle Matsuda, Will Tobias, Jun-Ru Li, Giacomo Valtolina and Luigi De Marco. Kyle’s optics knowledge and happiness made lab a joy to work in. Will’s electronics knowledge and guidance with projects saved me many times. Jun-Ru’s ability to explain what was going in lab at all times was incredible. Giacomo’s deep familiarity with the experiment and work ethic inspires me. I especially would like to thank Luigi who guided me so much in lab, helping me whenever I got stuck, explaining any physics concept in an understandable way, and even coming in to lab on multiple weekends to help me finish this project. His efforts truly made this project successful and he’s the best lab pong doubles partner anyone could ask for.

I would like to thank my friends all over the country, especially those who helped me survive vi high school and college. I wouldn’t be where I am today without all of you. Thank you especially to Albert H. W. Jiang and Caitlin Steele for always being there for me even in my darkest times.

Lastly, I would like to thank my family for their constant support me. Their sacrifices changed the course of my life for the better from an early age. vii

Contents

Chapter

1 Introduction 1

2 Theory, apparatus and motivation 3

2.1 Atomic structure ...... 3

2.2 Bosons and fermions ...... 4

2.3 Blasting atoms ...... 4

2.3.1 Rubidium D2 line ...... 4

2.4 Optical pumping theory ...... 6

2.4.1 Hyperfine splitting ...... 6

2.4.2 Transition strengths ...... 9

2.4.3 Transition linewidths ...... 10

2.5 Laser systems & cooling process ...... 10

2.5.1 Magneto-optical trap and molasses ...... 13

2.5.2 Plugged quadrupole and magnetic evaporation ...... 16

2.6 Optical evaporation ...... 17

2.7 Molecule production ...... 17

2.7.1 Magneto-association ...... 17

2.7.2 STIRAP ...... 18

2.7.3 Imaging molecules ...... 18 viii

2.7.4 Quantum degeneracy ...... 18

2.8 Molecule reactions ...... 19

2.8.1 KRb + KRb ...... 19

2.8.2 Molecule-atom reactions ...... 20

3 Laser system 22

3.1 Laser locking ...... 22

3.1.1 Laser details ...... 22

3.1.2 Temperature control ...... 23

3.1.3 Locking method ...... 24

3.1.4 Physical setup ...... 24

3.1.5 Electronics: locking ...... 28

4 Results 29

4.1 Lineshape of KRb ...... 29

4.2 Lifetime measurement ...... 31

4.3 Molecule number as a function of pulse time ...... 32

4.4 Lineshape on Rb ...... 33

5 Conclusion 35

5.1 Future directions ...... 35

Bibliography 36

Appendix

A Jupyter notebook for transition frequencies, strengths, and linewidths 38 ix

Figures

Figure

2.1 Rb D2 line data ...... 5

2.2 Ground-state Zeeman shifts ...... 8

2.3 Excited-state Zeeman shifts ...... 8

2.4 Relative transition strengths ...... 9

2.5 Relative transition strengths against frequency ...... 11

2.6 Transition linewidths ...... 12

2.7 Experiment steps ...... 14

2.8 MOT setup ...... 15

2.9 KRb donut ...... 21

3.1 Laser linewidth ...... 23

3.2 TEC replacement ...... 25

3.3 Experimental setup of high-field Rb laser ...... 26

3.4 Light to science cell ...... 27

4.1 Molecule lineshape ...... 30

4.2 Molecule lifetime ...... 31

4.3 Pumping over time ...... 32

4.4 Rubidium lineshape ...... 33

4.5 Rubidium lineshape (low power) ...... 34 Chapter 1

Introduction

In recent years, researchers have been increasingly interested in ultracold gases of molecules.

Progress has been achieved in creating ever denser and colder gases with the end goal of achiev- ing quantum degeneracy [1]. The motivation for this work is clear: ultracold, degenerate polar molecules can provide new insights into the behavior of particles in the quantum regime. They are ideal systems for tests of many branches of quantum mechanics including quantum information science [2], ultracold chemistry [1], quantum simulation [3], and low-dimension dynamics [1].

Polar molecules have stronger long-range dipolar interactions and more degrees of freedom compared to atoms. The added complexity of molecules over atoms means existing cooling tech- niques must be adapted and new ones created. Direct laser cooling of molecules has been demon- strated for certain, specially-chosen molecules but the phase-space density is nowhere near that needed for degeneracy [4]. An alternate method, used in our lab, continues work done on ultracold atoms by associating them into weakly-bound Feshbach molecules and then coherently transferring them to the molecular ground state. Recently, in a huge milestone and technological achievement, the first degenerate gas of polar molecules was created at JILA using ultracold potassium-rubidium

[5].

Production of increasingly degenerate gases is limited by inefficiencies throughout the cooling and molecule creation process. For example, the conversion efficiency from atoms to molecules is less than 50%, leaving a significant number of K and Rb atoms in the trap that destroy molecules quickly, limiting the time for experiments. The two atomic species will react with KRb, causing 2 heating and molecule loss. To blast away excess K, a resonant light pulse is used on an optical cycling transition. However, Rb atoms are not in a state with such a cycling transition, and so it is necessary to change the atomic hyperfine state to remove them. To accomplish this, we have typically used a series of microwave adiabatic rapid passages (ARPs), each followed by a pulse of resonant light, to remove a majority of trapped Rb. The time duration and non-unity efficiency of the ARPs + blast sequence was theorized by our lab as a contributor to molecule number

fluctuation between experimental runs. To improve molecule number and stability, a laser system was constructed to optically pump Rb into a state from which resonant light is used to complete

Rb removal. A frequency shift from the zero-field transition on the D2 was calculated to account for the high magnetic fields at which atomic removal occur.

This thesis details our first attempt at optically pumping Rb at high magnetic fields. In

Chapter 2, I give a motivation for my work along with a brief overview of atomic physics, followed by a discussion on the theoretical basis for high-field optical pumping of Rb. I then introduce the experimental apparatus used to create 40K87Rb, highlighting the cooling and molecule production steps to which my laser applies. In Chapter 3, I explain the setup of my laser system, detail the locking mechanism and discuss issues encountered along the path of construction. In Chapter 4, I present the results of my work on laser lineshape and molecule lifetime, discussing differences with the current method of numerous ARPs. In Chapter 5, I conclude the work shown in this thesis and propose future steps for high-field optical pumping. Chapter 2

Theory, apparatus and motivation

This chapter describes atomic structure before examining the D2 line for Rb. Then I discuss cooling steps in the KRb experiment used to create ultracold atomic gases. I touch on the lasers involved in each step, focusing on those used in Rb cooling. Lastly, I examine the two-step molecule process and conclude the chapter by examining where improvements can be made to increase molecule number and stability.

2.1 Atomic structure

Fine and hyperfine structure come from coupling of different angular momenta from the electron and nucleus. L is electron orbital angular momentum while S is spin angular momentum.

Therefore J, the total electron angular momentum, is given by

J = L + S , (2.1) where

L S J L + S. (2.2) | − | ≤ ≤ The projection m can be any value in integer steps between J and J. Total nuclear angular J − momentum, I, added to J gives the total atomic angular momentum:

F = J + I , (2.3) where

J I F J + I. (2.4) | − | ≤ ≤ 4

87 2 For ground-state Rb, 5 S1/2, J = 1/2 and I = 3/2, meaning F = 1 or F = 2. State notation

2S+1 is LJ with S, P referring to L = 0, L = 1 respectively. The D2 line for Rb refers to the

52S 52P transition. F represents excited states. For 52P , F can be 0, 1, 2 or 3. 1/2 −→ 3/2 0 3/2 0

2.2 Bosons and fermions

Bosons are particles with integer spin (0, 1, 2...) such as photons and hydrogen atoms. They obey Bose-Einstein statistics [6]. At ultracold temperatures, multiple identical bosons can occupy the same state leading to a Bose-Einstein condensate [7]. 87Rb is a boson.

Fermions are particles with half odd integer spin (1/2, 3/2, 5/2...) such as protons, neutrons and electrons. They obey Fermi-Dirac statistics [6]. The wavefunctions of identical fermions are antisymmetric under particle exchange meaning they cannot occupy the same state. This idea is the Pauli exclusion principle [7]. 40K is a fermion. 40K87Rb is fermionic since it is composed of a fermion and boson.

2.3 Blasting atoms

To do experiments on KRb, there must an undetectable number of atoms in the trap. Yet due to the experimental inefficiencies, more atoms are left than molecules produced. Both K and Rb will destroy KRb on the millisecond timescale [8]. To prevent molecule loss, both atomic species are blasted away, out of the trap. Potassium is removed with resonant light on the cycling transition

F = 9/2, m = 9/2 F = 11/2, m = 11/2 . When we dissociate the molecules, Rb is | F − i → | 0 F0 − i not in a state with a cycling transition so it requires two steps to be blasted out of the trap. To understand why requires looking in the energy levels of Rb.

2.3.1 Rubidium D2 line

2 2 The Rb D2 line is the transition from the 5 S1/2 to 5 P3/2 state. 5

Figure 2.1: Rb D2 line data. Figure taken from [9].

Rubidium must be in the F = 1, m = 1 state for magneto-association and so unconverted | F i Rb atoms are left in that state. Since photons are spin 1 particles, a transition can only change m by 0, 1. Therefore, F = 3, m = 3 is only accessible from the 2, 2 state, making this F ± | 0 F0 i | i a cycling transition on which Rb can be blasted. The direct method of getting to the necessary ground state of 2, 2 is a series of spin-flip, radio-frequency (RF) sweeps centered near 6.8GHz. | i This type of sweep is called adiabatic rapid passage (ARP). Four RF ARPs are used sequentially, 6 each with around 95% efficiency. Each ARP takes about 750 µs, time during which molecules can be destroyed. Additionally, the ARP’s non-unity efficiency means the end number of molecules is lower and less stable than it could be. Lastly, RF pulses and sweeps interfere with nearby electronics, causing issues like magnetic field instability.

2.4 Optical pumping theory

Optical pumping presents a quicker, more efficient method of blasting Rb than radio-frequency

(RF) ARPs. This alternative technique involves pumping from the 1, 1 to F = 2, m = 2 state | i | 0 F0 i which spontaneously decays quickly to the 2, 2 state. The Rb repump laser, discussed later in | i this chapter, hits this transition but at a vanishing magnetic field. After molecule production, the magnetic field is 545.6 G. Magnetic fields cause Zeeman splitting, shifting the energy levels enough to require construction of a new laser system for optical pumping at high magnetic fields.

Optical pumping requires knowing the frequency of a transition along with decay pathways.

Magnetic fields shift the energy levels of the hyperfine structure. To calculate this shift, the hyperfine and magnetic Hamiltonians must be diagonalized.

2.4.1 Hyperfine splitting

The hyperfine structure Hamiltonian is [10]

3(I J)2 + 3 I J I(I + 1)J(J + 1) H = A I J + B · 2 · − , (2.5) hfs hfs · hfs 2I(2I 1)J(2J 1) − − giving energy shifts of

1 3 K(K + 1) 2I(I + 1)J(J + 1) ∆E = A K + B 2 − , (2.6) hfs 2 hfs hfs 2I(2I 1)2J(2J 1) − − where Ahfs is the magnetic dipole constant, Bhfs is the electric quadrupole constant and

K = F (F + 1) I(I + 1) J(J + 1) . (2.7) − − 7

This Hamiltonian is diagonal in the coupled basis. The Hamiltonian from the magnetic field interactions in the z-direction is [10]

HZ = µB (gI Iz + gJ Jz) Bz . (2.8)

The Feshbach resonance for KRb lies in the intermediate field regime meaning the total Hamilto- nian, Hhfs + HB, must be diagonalized since neither can be treated as a perturbation of the other.

When J = 1/2, H can be solved analytically, giving rise to the Breit-Rabi formula [11],

1/2 ∆Ehfs ∆Ehfs 4mx 2 E J=1/2, m , I, m = + gI µBmB 1 + + x , (2.9) | J I i −2(2I + 1) ± 2 2I + 1   where m = m m and I ± J (g g ) µ B x = J − I B . (2.10) ∆Ehfs

Otherwise the total Hamiltonian must be diagonalized numerically. The Wigner-Eckart theorem, after much simplification, gives the Jz matrix elements in the coupled basis [12]

F +F +J m +I+1 F, m J F 0, m0 = ( 1) 0 − F J(J + 1)(2J + 1) (2F + 1) (2F + 1) F | z| F − 0 p p F 1 F 0 J 0 F 0 I , (2.11) ×     mF 0 mF0  FJ 1   −      where is the 3-j symbol and is the 6-j symbol. The Iz matrix elements can be found by ! () swapping I and J in Eq. 2.11.

Lastly, to explain the ordering of the states in Figure 2.2, note that HB is a perturbation to

Hhfs when the magnetic field is small giving

∆E F, m = µBgF mF Bz , (2.12) | F i where gF is the hyperfine Land´eg-factor. It can be approximated as

F (F + 1) I(I + 1) + J(J + 1) g g − . (2.13) F ' J 2F (F + 1) 8

2 Figure 2.2: Zeeman shifts of ground state 5 S1/2. Produced from code in appendix.

2 Figure 2.3: Zeeman shifts of 5 P3/2 excited state. Produced from code in appendix. 9

Therefore, for F = 1 in both the ground state and excited state on the D2 line, gF has a negative

2 sign. The effect of this sign flip is represented visually for 5 S1/2 on Figure 2.2 and explains why

1, 1 is the lowest energy ground state. | i The frequency increase between the 1, 1 to F = 2, m = 2 transition at no magnetic field | i | 0 F0 i versus at 545.6 G is calculated to be 1.07 GHz (Appendix A). The total frequency of the laser is then supposed to be near 384.23584 THz.

2.4.2 Transition strengths

The viability of high-field optical pumping is based on Rb in the F = 2, m = 2 state | 0 F0 i falling into the 2, 2 state instead of any other state. The probabilities of decay paths are given | i by relative optical transition strengths.

+ Figure 2.4: Relative transition strengths for σ , π, and σ− transitions of the D2 line. States for both axes are ranked from lowest to highest energy for B = 545.6G. Figure produced from code in appendix.

Figure 2.4 gives a visual representation for transition strengths on the D2 line. For example, the cycling transition used for imaging and blasting is seen as a dark red rectangle in the top right corner of the first panel, which shows the F = 3, m = 3 will only decay to the F = 2, m = 2 | 0 F0 i | F i state. The key rectangles are those for excited state 11 ( 2, 2 0) that show three possible decays. | i 10

Emission into the 1, 1 state is not an issue because the laser constantly re-excites these Rb atoms. | i There is a faint rectangle representing a small chance of Rb falling to ground state 6 ( 2, 1 ), which | i is not desirable since these atoms cannot be pumped and will react with KRb. The right-most rectangle for excited state 11 shows a π transition down to the 2, 2 state which is the ideal since | i these atoms posses a cycling transition and can be blasted away by imaging light. The top panel in Figure 2.5 shows the two σ+ transitions from the 1, 1 state are separated by over 1 GHz, | i which is much more separation than is needed to selectively pump into just one excited state.

From the relative transition strengths, the probability of falling into the 2, 2 state is found to be | i 97.78%.

2.4.3 Transition linewidths

To minimize the pulse time needed for optical pumping, the laser linewidth should be smaller than the transition linewidth. The laser linewidth was measured to be 1.08MHz and is described in the next chapter. The 1, 1 2, 2 0 transition has a linewidth of 2.00MHz, calculated in | i → | i Appendix A. These two linewidths are close enough together for optical pumping to be successful.

The other important linewidth value is for the 2, 2 0 2, 2 transition. It was calculated | i → | i to be 3.97MHz. The inverse linewidth gives the decay rate, in this case 40 ns, which is over three orders of magnitude shorter than the duration of one RF ARP.

2.5 Laser systems & cooling process

Creating ultracold KRb through association of ultracold 40K and 87Rb builds directly on previous work done at JILA on both atomic species. The first Bose-Einstein condensate (BEC) and degenerate Fermi gas were made at JILA by cooling 87Rb and 40K, respectively [13, 14]. Indeed before making KRb, the first step is cooling its constituent atoms to ultracold temperatures, which is accomplished with a sequence of steps typical in atomic, molecular, and optical (AMO) physics.

For this experiment, we use a dual-species magneto-optical trap (MOT), followed by a compressed

MOT. Then sub-Doppler cooling is achieved for both atomic gases with optical molasses. The 11

+ Figure 2.5: Relative transition strengths for σ , π, and σ− transitions of the D2 line for different transition energies for (A) the 1, 1 state and (B) the 2, 2 0 state. The x-axes are both the same | i | i shifted transition energy. Figure produced from code in appendix A. 12

+ Figure 2.6: Linewidths for σ , π, and σ− transitions of the D2 line for different transition energies for (A) the 1, 1 state and (B) the 2, 2 0 state. The x-axes are both the same shifted transition | i | i energy. Figure produced from code in appendix A. 13 atoms are then trapped in a magnetic field that is used to spatially transport the atoms to a high-vacuum cell. From here, the Rb temperature is lowered further with magnetic evaporation while K cools sympathetically through collisions with Rb. The quadrupole trap is turned off while a crossed optical dipole trap (xODT) and magnetic bias field are turned on, keeping the colder atoms. Further increases in phase-space density come from optical evaporation. From this cold mixture of K and Rb, molecules can finally be created. Figure 2.7 shows this general sequence along with the temperature change through each step.

2.5.1 Magneto-optical trap and molasses

The idea behind a magneto-optical trap (MOT) was so revolutionary that it won the 1997

Nobel Prize [15]. MOTs are generally the first step in creating a cold gas of alkali atoms. There are two interconnected parts of a MOT: magnetic fields and optical cooling. To cool down an atomic gas, six red-detuned beams are used [7]. An atom stationary in the lab frame is in a dark state and therefore doesn’t absorb any photons. Atoms moving radially outward have their transitions red- shifted by the Doppler effect into resonance with the red-detuned laser opposing the atom’s motion.

Spontaneous emission occurs rapidly in a uniformly random direction. This process reduces the mean kinetic energy of the atoms down to a limit called the Doppler temperature [7].

Doppler cooling does not confine atomic gases [7]. That comes from the addition of a magnetic

field created by two anti-Helmholtz coils, shown in Figure 2.8. At the center of the trap is a point with zero magnetic field from which the field strength is approximately linear as a function of position, resulting in splitting of hyperfine and fine sublevels [16]. Additionally, the red-detuned beams are circularly polarized to match the type of transition whose energy level has dropped from the zero-field energy.

Maneto-optical traps for Rb often use the F = 2 F = 3 transition [17]. Normally Rb → 0 atoms in the excited state of F 0 = 3 decay back into the F = 2 state, which allows for further cooling. However, there is a small chance of dropping into the F = 1 state, so a repump laser is 14

Figure 2.7: Number of K, Rb, and KRb are shown on the first panel. The second panel shows the temperature over a run of the experiment. Note the x-axis (time) is not to scale. Figure taken from supplementary material of [5]. 15

Figure 2.8: Magneto-optical trap example for two-level atom with ground state J = 0 and excited state J = 1. (a) Two anti-Helmholtz coils generate a magnetic field (blue lines). Six beams cross at the center. (b) Energy levels of mJ0 states along with the ground state shown centered on the + zero-field. A red-detuned beam (solid red line) crosses at two points, one for σ and one for σ− polarization. Figure taken from [16]. 16 used to continually excite to these atoms into the F 0 = 2 state, which falls into the F = 2 state

[18].

Both atomic species, K and Rb, are trapped in the same space inside an atomic vapor cell

7 which is under vacuum down to around 10− torr [5]. The exponential fill time for this dual-species

MOT is about 2.5 seconds resulting in around N = 108 and N = 2 109 for a typical run [5]. K Rb × After the MOT is loaded, the magnetic gradient is increased to form a compressed MOT which gives higher phase-space density [19].

After the MOT is filled, sub-Doppler cooling is achieved with optical molasses. Potassium is cooled with a Λ-enhanced gray molasses on the D1 line for 10 ms [5]. For Rubidium, a bright molasses is used for 2 ms followed by a Λ-enhanced gray molasses for 8 ms, both operating on the

D2 line [5]. At this point, the K and Rb clouds are at 20 and 10 µK respectively. After laser cooling, the distribution of states is random for each atom so a bias field of 30 G is used to allow for optical pumping of Rb into the 2, 2 state and K into the 9/2, 9/2 state [5]. To prepare the atoms for | i | i spatial transport, a quadrupole field is turned on, which adiabatically compresses the cloud. Then the anti-Helmholtz coils move, with the atoms, about a meter horizontally into a lower pressure, science cell. Differential pumping results in pressure of 10 11 torr [5]. From here, further cooling ∼ − steps can be taken.

2.5.2 Plugged quadrupole and magnetic evaporation

Evaporation is a process in which the hottest atoms are released from the trap, simultaneously lowering the number of particles and temperature. Two different types of evaporation are used in this experiment to bring the temperature down to the ultracold, sub-µK regime: twenty seconds of plugged quadrupole evaporation is followed by seven seconds of optical evaporation [5].

Quadrupole evaporation uses anti-Helmholtz coils and a plug beam [7]. It is performed only on Rb and causes most of these atoms to be lost, which explains the large initial ratio of Rb to K after magnetic transport [5]. The magnetic field is zero at the center of the trap so Rb mF sublevels are degenerate [7]. This means spin-flip transitions can occur to untrapped states, a process called 17

Majorana loss [20]. To prevent rapid atomic loss, a blue-detuned plug laser, which is repulsive for atoms, is used that prevents atoms from reaching the center of the trap. Here, the lifetime of the trapped atoms is around 150 s, well beyond what is needed [5]. An RF knife near 6.8 GHz is used to selectively drive the hot atoms into the anti-trapped high-field seeking 1, 1 state [5]. During | i plugged quadrupole evaporation, K is sympathetically cooled through collisions with Rb [21]. At this point during a typical run, the temperature is 4µK for both species, with approximately ∼ 6 106 of each [5]. ×

2.6 Optical evaporation

Following magnetic evaporation is optical evaporation which uses two elliptical beams to form a crossed optical dipole trap (xODT). The quadrupole trap and plug beam are turned off after the xODT is made, followed by putting the bias field at 30 G which causes Zeeman splitting [5]. Optical evaporation utilizes the same principles as magnetic evaporation: hotter atoms leave the trap. The intensities of the crossed beams are ramped down exponentially, lowering the trap depth, keeping only colder atoms trapped [22]. This step is the last in the cooling sequence before molecule production. At this point, there are often 105 Rb and 10 times more K, with temperatures in ∼ ∼ the tens or hundreds of nK [5]. Lower temperatures are achieved with more evaporation, lowering the final atom counts.

2.7 Molecule production

Producing KRb molecules consists of two steps: magneto-association and STIRAP.

2.7.1 Magneto-association

Initial molecule production is achieved through magneto-association, which involves manipu- lation of magnetic fields to move adiabatically from atomic states to a molecular state. Specifically, we ramp the magnetic field down from 556 G to 545.6 G through a Fano-Feshbach resonance at

546.6 G, where the atom-molecule switch occurs. The ramp is done slowly enough, 3 ms in this 18 case, to stay in an eigenstate of the system so that the adiabatic theorem is satisfied [5]. Magneto- association only converts 10-50% of Rb into weakly-bound Feshbach molecules depending on ratio of K to Rb [5].

2.7.2 STIRAP

KRb Feshbach molecules are weakly bound [18]. Therefore, they are quickly converted into rotational-vibrational (rovibrational) ground-state molecules using stimulated Raman adiabatic passage (STIRAP) [5]. This two-photon process removes 6000 K of binding energy for each molecule without affecting temperature [18]. STIRAP is typically 85% efficient.

2.7.3 Imaging molecules

For imaging, STIRAP is done in the reverse direction to remake Feshbach molecules, which can be imaged using resonant light but their absorption cross section is smaller than that of their component atoms. Therefore, we tend to ramp back up through the Feshbach resonance, which dissociates the molecules, and image the atoms themselves. For K, imaging takes place on the

9/2, 9/2 11/2, 11/2 cycling transition. Rb is imaged on the 2, 2 3, 3 cycling | − i → | − i0 | i → | i0 transition. To put the Rb in an imageable state, an ARP is used.

2.7.4 Quantum degeneracy

I note earlier that a degenerate Fermi gas of ultracold polar molecules was first achieved at JILA in 2018, specifically in the KRb lab. Key to understanding degeneracy is the Fermi temperature, written as

~ω 1/3 TF = (6N) , (2.14) kB where N is the number of particles and ω is the harmonic trapping frequency [23]. A measure of degeneracy is given by the temperature of the system divided by Fermi temperature, T/TF.

The lower this value is, the more degenerate a system is. Intuitively, this can be understood as degeneracy is reached when the system is very cold and very dense. Since the Pauli exclusion 19 principle prevents fermions from occupying the same state, the molecules’ energies stack up. There are two ways to make an ultracold gas more degenerate: decrease the temperature or increase the

Fermi temperature. The simplest way to accomplish the latter is increasing the number of molecules while maintaining the same trap size. This method of increasing degeneracy is why blasting Rb is important. Additionally, when making ultracold KRb, colder temperatures can be achieved by evaporating atomic mixtures more, but this limits the number of molecules that can be made [5].

Therefore for the more degenerate KRb gases, it is even more important to remove Rb quickly because each molecule makes up a larger percentage of total number.

2.8 Molecule reactions

Ultracold chemical reactions seem unintuitive classically due to the low temperatures. But dynamics in this temperature regime run according to quantum statistics [8]. The de Broglie wavelength increases as the cloud gets colder, largely determining how particles interact. Reactions take place at large intermolecular distances due to spacial overlap of the wavefunctions. Loss of molecules can occur through reactions between a pair of them or from reactions with atoms. The previous version of the KRb experiment, generation I, probed these reactions rates [8].

2.8.1 KRb + KRb

Molecule-molecule reactions are the only loss mechanism when no atoms remain in the trap.

The reaction KRb + KRb K + Rb is the most common reaction and releases 10 cm 1 of → 2 2 ∼ − kinetic energy [8]. The number density of KRb, n, over time is fitted to the equation

dn 3 n dT = βn2 , (2.15) dt − − 2 T dt where β is the two-body loss rate [8]. The second term represents heating in the form of anti- evaporation [5].

K Rb K KRb has 36 hyperfine states since I = 4 and I = 3/2, giving nine mI projections and four from mRb. STIRAP produces molecules in the excited, singlet spin state mK = 4, mRb = 1/2 . I I − I

20

The absolute ground state is 4, 3/2 and is reached by two π pulses, using an intermediate, |− i rotationally excited state in which nuclear spins can be flipped [24]. The loss rate for both states, yielded from the slope of a β (rate coefficient) versus T linear fit, are similar at around 1.2(3) × 5 3 1 1 10− cm s− K− [8].

2.8.2 Molecule-atom reactions

Any molecule-atom reactions are detrimental because they limit the number of molecules on which experiments can be performed. The reaction K + KRb K + Rb is exothermic while → 2 Rb + KRb Rb + K is endothermic and therefore is in principle forbidden [8]. In the previous → 2 generation of KRb, both reactions were examined where the number of atoms heavily outweighed that of the molecules, so that atom number density remained essentially constant. Molecule number,

Nmolecule, can be written as

d N = β N n (2.16) dt molecule − × molecule × atom where β is the inelastic rate coefficient and natom is the atomic density [3]. Since the atom to molecule ratio is high, this differential equation can be solved approximately giving

N N exp( β n t) , (2.17) molecule ' 0 − × atom × where N0 is the initial number of molecules. From this equation, β can be found by curve-fitting.

Molecules in the 4, 3/2 state are lost much quicker in the presence of K than Rb [8]; the |− i inelastic coefficient for K + KRb was found to be 1.7(3) 10 10 cm3 s 1 while that of Rb + KRb × − − was 0.13(4) 10 10 cm3 s 1. This gave a molecule lifetime of 7(2) ms when K was in excess [8]. × − − The reaction mechanism for KRb loss in the presence of Rb is not well understood [8]. A possible pathway is three-body loss (Rb + Rb + KRb) [8]. The reaction rate jumps to near that of

K + KRb if either Rb or KRb are not in their lowest-energy internal states [8]. Generally our lab performs experiments on KRb in the 4, 1/2 state. Therefore, getting rid of all atoms is critical |− i once the molecules are made since the lifetime of the molecules by themselves ( 2-3 s) is much ∼ longer than when there are excess atoms ( 5-10 ms) [8]. ∼ 21

Figure 2.9: Image of a KRb cloud with a hole in the center.

Figure 2.9 shows an image of the molecules when Rb atoms are not removed quickly enough.

In this case, a Rb BEC in the center of the trap was not fully blasted out in time and reacted with

KRb molecules. As previously mentioned, for Rb removal, four ARPs are used over a period of around 4 ms, a decent chunk of the molecule lifetime when many atoms are present. The non-unity efficiency of the ARPs present a window for improvement; if Rb can be optically pumped in micro- or even nanoseconds, the number of molecules should increase and be more stable. Additionally, reductions in RF usage contribute to stability in electronics for the rest of the experiment. Chapter 3

Laser system

3.1 Laser locking

Lasers without feedback loops will drift in frequency over time due to temperature and current

fluctuations. Temperature changes are mitigated by a thermoelectric controller (TEC) with PID feedback. Locking a laser to a specific frequency requires a reference and a current feedback loop.

For the high-field optical pumping laser (HF OP), the reference is the repump laser. That laser’s reference is Rb atoms in a heated vapor cell. For details on the repump laser including how it is locked, I refer the reader to Steven Moses’ 2016 PhD thesis [18].

3.1.1 Laser details

The HF OP laser is a 780 nm Photodigm DBR in a Mercury TOSA package in a heat sink mount. It is run at 72 mA and near 22.061 C. The laser has a linewidth below 1.08 ∼ ◦ MHz, well within the typical and useful range. The linewidth measurement was taken using the self-heterodyne technique where two paths of a laser, one travelling 100 m in a fiber in this case and the other through an acousto-optic modulator (AOM), are recombined on a photodiode [25].

This method operates on the principle that the delayed path distance is longer than the coherence length, meaning the photodiode output is the self-convolution of the laser output [25]. Since lasers output a Lorentzian spectrum, the self-convolution is also Lorentzian distribution. From a fit, the distribution width can be extracted, which is a multiple of √2 larger than the linewidth. 23

Figure 3.1: Laser linewidth. The center occurs at 80 MHz because one path goes through an AOM.

3.1.2 Temperature control

The TEC for the Mercury TOSA laser is mounted directly on the diode but was broken in this instance. To fix this issue, a square section of the heat sink metal was removed directly below the diode. The missing metal was replaced with two pieces of aluminum, a top and bottom piece, with room in-between for a 10 mm 10 mm TEC as shown in Figure 3.2. The top piece has ∼ × a slot cut to exactly fit the diode so that it rests directly on the TEC. A screw that presses the diode into the TEC from above along with four more on the corners of the top piece create a stable foundation for the diode.

After several attempts adjusting TEC placement and tightness along with PID adjustment, the temperature was controlled down to an oscillation with an amplitude of 10 mK over several seconds. The response time to temperature changes from manual current adjustments was slow, 24 likely due to the heat capacity of the added aluminum. This made locking difficult for some time since there was a wait period each time the lock was missed. Strangely, after four months without the diode being physically adjusted, the temperature control improved greatly with the oscillation amplitude decreasing to sub-mK levels and response time reducing substantially as well.

One possibility is that the thermal paste spread over time under pressure which increased thermal conductivity.

3.1.3 Locking method

The HF OP laser is locked by creating a beatnote with the repump laser onto a photodiode.

The interference of the two lasers creates a beatnote: a sine wave with a frequency of the two inputs’ difference. The frequency of the beatnote is what must be kept at 1.07 GHz for successful pumping.

3.1.4 Physical setup

The laser system (Figure 3.3) involves optics that collimate, isolate, split, shift, and collect the beam. The light coming out of the laser spreads vertically but is collimated horizontally so two cylindrical lenses are used to make the beam cross section circular. Then comes a free-space isolator, which uses the Faraday effect and two polarizing beam splitters (PBS) to prevent any reflected light from reflecting back to the laser followed by a λ/2 waveplate, changes the beam polarization. Next, a PBS separates s- and p-polarization, creating two different paths. One path goes through a shutter and an AOM, which are used to quickly block and unblock the beam, before being focused into a fiber that routes light to the experiment. The AOM shifts the beam frequency by integer multiples of 80 MHz along with changing its propagation angle. For this setup, the first order was used, increasing light frequency by 80 MHz. The other beam path is used to create the beatnote. It is overlapped with light from the repump laser in a beam splitter before being focused onto a photodiode. 25

Figure 3.2: TEC replacement. The top piece of aluminum screws down on all four corners into the heat sink mount. 26

(a) Laser setup and electronic feedback. Red lines are beams and blue lines are wires.

(b) Physical setup

Figure 3.3: High-field optical pumping laser. 27

Figure 3.4: Fiber and simplified optics setup. The HF OP light is combined with imaging light in a fiber splitter. A λ/4 waveplate circularly polarizes both beams because they need to be σ+ to increase mF . The first two mirrors in the path represent those used for alignment. The last mirror represents the 45o mirror that reflects the light downwards, switching the propagation direction from in the x-y plane to the negativez ˆ-direction. 28

3.1.5 Electronics: locking

The beatnote produced by the photodiode is shifted down to tens of MHz by mixing with the output of a voltage-controlled oscillator (VCO). The signal is changed into a voltage using a frequency-to-voltage converter (F2V). The output of the F2V is fed into a loop filter, which uses a gain curve to output a signal pushing the laser closer to the ideal frequency of 384.23584 THz. Chapter 4

Results

The aim of this research was to improve KRb molecule number and stability. A transition for high-field optical pumping near the predicted frequency was found. The key value that determines whether this method was an improvement is the molecule lifetime in the form of reaction rate. No improvement by optical pumping was found over the RF ARPs in molecule lifetime.

4.1 Lineshape of KRb

From a Lorentzian fit to a lineshape measurement, the peak frequency can be extracted.

Using the HF OP laser on the molecules, the peak beatnote frequency was found to be 1057(1)

MHz. Small shifts from the predicted peak can be partly attributed to both the experimental side, such as magnetic field uncertainties, and the electronics side of the laser. The peak molecule number was close to the typical number produced using ARPs. The laser power for this measurement was

1 µW with a pulse time of 100 µs. The hold time was 100 ms.

The measurement in Figure 4.1 shows the HF OP laser is hitting the correct transition since nearly all the molecules would be destroyed after 100 ms if no Rb were removed. The peak molecule number being similar for both the ARPs and HF OP shows that they are both blasting approximately equal numbers of Rb after the ground-state molecules are made. To get a better understanding of how efficient and quick both methods are, molecule lifetimes were measured as shown in the next subsection. 30

Figure 4.1: KRb number as a function of beatnote frequency. 31

4.2 Lifetime measurement

The lifetime of the molecules depends on the number of atoms left. Fewer Rb atoms in the trap mean the molecules live longer. Multiple lifetimes were measured with slight variations in laser frequency with the maximum lifetime being consistent with that of the ARPs.

Figure 4.2: Ground-state molecule density as a function of time. Fitting is done with exponential curves. The frequency for each HF OP colored set of points is beatnote frequency.

In Figure 4.2, the red points are density measurements from the ARPs + blast sequence we typically use. For these, a lifetime of 1.16(6) 10 12 cm3/s was extracted. The other three × − measurements were taken using high-field optical pumping instead. The blue, green, and pink point correspond to beatnote frequencies of 1166 MHz, 1072 MHz, and 1056 MHz, respectively. In order, these correspond to lifetimes of 1.3(3) 10 12 cm3/s, 1.3(1) 10 12 cm3/s, and 1.4(2) 10 12 cm3/s. × − × − × − This means that high-field optical pumping works just as well as the ARPs at removing Rb from 32 the trap since the molecule have the same reaction rate for both methods.

Realistically, this result ends the idea of switching the Rb blasting over to high-field optical pumping from RF, since the difference in lifetime and number of KRb produced is statistically insignificant for the two methods. In addition, lasers can go unlocked while ARPs use digital synthesizers that do not need a reference. Possible reasons for the lack of lifetime improvement include Rb that fall into the 2, 1 state during pumping, since these would react with KRb. | i

4.3 Molecule number as a function of pulse time

To see how quickly the HF OP blasted Rb, a series of measurements were taken at different pulse times with a power of 1 µW. Quicker blasting should result in higher molecule numbers since fewer Rb + KRb reactions are able to occur. Molecule number saturation for a beatnote frequency of 1072 MHz is reached in 250 µs as shown by the purple points in Figure 4.3. This time is much ∼ shorter than the duration of the four ARPs + blasts. The red point is an ARP measurement and placed as a reference only.

Figure 4.3: Number of KRb after a hold time of 100 ms as a function of pulse time with two different laser powers. 33

4.4 Lineshape on Rb

In addition to measuring the molecule loss, we can directly measure the number of Rb atoms in the 2, 2 state after optical pumping. | i

Figure 4.4: Number of Rb as the HF OP frequency offset is changed.

A lineshape on Rb atoms at moderate laser powers should give a Lorentzian centered near

1070 MHz. Rather, in Figure 4.4, there are two Lorentzians for each power with dips near predicted peak frequency of 1070 MHz. This result was seen multiple times when the power was in the µW range. Lowering the power to the nW range produced a single Lorentzian centered close to the predicted peak frequency. 34

Figure 4.5: Number of Rb as the beatnote frequency is changed. Points were fit with a Lorentzian with a peak frequency of 1090 MHz. The HF OP laser power was in the tens of nW.

Though the curve in Figure 4.5 has the right shape, the peak number of Rubidium was 1/3 ∼ the value it should have been. All of these lineshapes on Rubidium were taken before testing the laser with a molecule measurement (Fig. 4.2). While the HF OP laser did not work as predicted for imaging Rb, it was successful in preventing KRb molecules from being destroyed by Rb atoms. Chapter 5

Conclusion

In this thesis, I have introduced the KRb generation II experimental apparatus, discussed ultracold atoms and molecules, explained the theory and motivation behind high-field optical pump- ing, and showed the new laser system I created. The goal of building a new system was to blast Rb quicker than a series of RF ARPs to increase the number of ground-state KRb molecules. While no considerable improvement was seen, success can be found in high-field optical pumping working alone. At some point or another, this method was going to be investigated as a way to produce more deeply degenerate molecular gases.

5.1 Future directions

Due to the low power required for high-field optical pumping, it is feasible to use an EOM on light from the repump laser. Also, there are other locations in the experimental sequences improvements can be made to increase the number of molecules created. As time goes on, molecular gases will surely be produced with ever-increasing degeneracy. Bibliography

[1] Lincoln D Carr, David DeMille, Roman V Krems, and Jun Ye. Cold and ultracold molecules: science, technology and applications. New Journal of Physics, 11(5):055049, 2009.

[2] SF Yelin, K Kirby, and Robin Cˆot´e. Schemes for robust quantum computation with polar molecules. Physical Review A, 74(5):050301, 2006.

[3] Klaus Osterloh, Nuria Barber´an,and Maciej Lewenstein. Strongly correlated states of ultracold rotating dipolar fermi gases. Physical review letters, 99(16):160403, 2007.

[4] Lo¨ıcAnderegg, Benjamin L Augenbraun, Yicheng Bao, Sean Burchesky, Lawrence W Cheuk, Wolfgang Ketterle, and John M Doyle. Laser cooling of optically trapped molecules. Nature Physics, 14(9):890–893, 2018.

[5] Luigi De Marco, Giacomo Valtolina, Kyle Matsuda, William G Tobias, Jacob P Covey, and Jun Ye. A degenerate fermi gas of polar molecules. Science, 363(6429):853–856, 2019.

[6] D.V. Schroeder. An Introduction to Thermal Physics. Always learning. Pearson Education, 2013. ISBN 9781292026213.

[7] D.H. McIntyre, C.A. Manogue, J. Tate, and Oregon State University. Quantum Mechanics: A Paradigms Approach. Always learning. Pearson, 2012. ISBN 9780321765796.

[8] S Ospelkaus, K-K Ni, D Wang, MHG De Miranda, B Neyenhuis, G Qu´em´ener,PS Juli- enne, JL Bohn, DS Jin, and J Ye. Quantum-state controlled chemical reactions of ultracold potassium-rubidium molecules. Science, 327(5967):853–857, 2010.

[9] Daniel A Steck. Rubidium 87 d line data, 2001.

[10] G.K. Woodgate. Elementary Atomic Structure. Oxford Science Publications. Clarendon Press, 1980. ISBN 9780198511564.

[11] Alan Corney. Atomic and laser spectroscopy. Clarendon Press Oxford, 1978.

[12] D.M. Brink and G.R. Satchler. Angular Momentum. Oxford Science Publications. Clarendon Press, 1993. ISBN 9780198517597.

[13] Mike H Anderson, Jason R Ensher, Michael R Matthews, Carl E Wieman, and Eric A Cornell. Observation of bose-einstein condensation in a dilute atomic vapor. Science, pages 198–201, 1995. 37

[14] CA Regal, Markus Greiner, and Deborah S Jin. Observation of resonance condensation of fermionic atom pairs. Physical Review Letters, 92(4):040403, 2004.

[15] R Srinivasan. 1997 nobel prize for physics: Laser cooling and trapping of atoms. Current Science, 74(2):106–110, 1998.

[16] Jabez J McClelland, Adam V Steele, B Knuffman, Kevin A Twedt, A Schwarzkopf, and Truman M Wilson. Bright focused ion beam sources based on laser-cooled atoms. Applied physics reviews, 3(1):011302, 2016.

[17] Sara Rosi, Alessia Burchianti, Stefano Conclave, Devang S Naik, Giacomo Roati, Chiara Fort, and Francesco Minardi. λ-enhanced grey molasses on the d 2 transition of rubidium-87 atoms. Scientific reports, 8(1):1–9, 2018.

[18] Steven Aaron Moses. A quantum gas of polar molecules in an optical lattice. PhD thesis, University of Colorado Boulder, 2016.

[19] Wolfgang Petrich, Michael H Anderson, Jason R Ensher, and Eric A Cornell. Behavior of atoms in a compressed magneto-optical trap. JOSA B, 11(8):1332–1335, 1994.

[20] DM Brink and CV Sukumar. Majorana spin-flip transitions in a magnetic trap. Physical Review A, 74(3):035401, 2006.

[21] JJ Zirbel, K-K Ni, S Ospelkaus, JP D’Incao, CE Wieman, J Ye, and DS Jin. Collisional stability of fermionic feshbach molecules. Physical review letters, 100(14):143201, 2008.

[22] Charles S Adams, Heun Jin Lee, Nir Davidson, Mark Kasevich, and Steven Chu. Evaporative cooling in a crossed dipole trap. Physical review letters, 74(18):3577, 1995.

[23] DA Butts and DS Rokhsar. Trapped fermi gases. Physical Review A, 55(6):4346, 1997.

[24] S Ospelkaus, K-K Ni, G Qu´em´ener,B Neyenhuis, D Wang, MHG De Miranda, JL Bohn, J Ye, and DS Jin. Controlling the hyperfine state of rovibronic ground-state polar molecules. Physical review letters, 104(3):030402, 2010.

[25] L Richter, H Mandelberg, M Kruger, and P McGrath. Linewidth determination from self-heterodyne measurements with subcoherence delay times. IEEE Journal of Quantum Electronics, 22(11):2070–2074, 1986.

[26] DS Naik and C Raman. Optically plugged quadrupole trap for bose-einstein condensates. Physical Review A, 71(3):033617, 2005. Appendix A

Jupyter notebook for transition frequencies, strengths, and linewidths

The attached code was written by Luigi De Marco, a postdoc in the KRb lab. I made small edits to the plotting parts of the code. Plots are commented out to prevent redundancy. 39

hyperfineShifts

March 31, 2020

1 Hyperfine & Zeeman Shifts in Rb D2 Transition

[30]: %matplotlib inline import numpy as np import matplotlib.pyplot as plt import matplotlib.patches as mpatches plt.rcParams.update({'font.size': 18})

plt.rcParams["figure.figsize"] = (12,10) plt.rcParams['font.size'] = 22

from wigner import * import const

1.0.1 Function Definitions and Matrix Elements

In this notebook, we work in the coupled basis F, m , which are simulatneous eigenstates of F 2 | F ⟩ and FZ . The F operator is given by

F = J + II, (1)

where J is the total electronic angular momentum (orbital plus spin) and I is the nuclear spin. [2]: def stateList(f): # Returns the list of F and MF states for a given list of F states mF = [] F = [] for k in f: mF += list(np.arange(-k,k+1,1)) F += [k]*(2*k+1) return F, mF

The hyperfine hamiltonian is given by

2 3 3(J I) + 2 (J I) I(I + 1)J(J + 1) Hhfs = AhfsJ I + Bhfs · · − , (2) · 2I(2I 1)J(2J 1) − − 1 40

and is diagonal in the coupled basis since

1 K J I F, m = (F (F + 1) J(J + 1) I(I + 1)) F, m = F, m . (3) · | F ⟩ 2 − − | F ⟩ 2 | F ⟩ The matrix elements of the hyperfine hamiltonian are therefore

3 K 2 K(K + 1) I(I + 1)J(J + 1) F mF Hhfs F ′m′ = Ahfs + Bhfs − δm ,m δF,F . (4) ⟨ | | F ⟩ 2 2I(2I 1)J(2J 1) F F′ ′ ( − − )

[3]: def hyperfineHamiltonian(F, I, J, A, B): h = []

for i in F: K = i*(i+1)-J*(J+1)-I*(I+1) x = A*K/2.0 if J > 0.5: x += B*(1.5*K*(K+1)-I*(I+1)*J*(J+1))/(2.0*I*(2.0*I-1)*J*(2*J-1)) h.append(x)

return np.diag(h)

The Zeeman Hamiltonian is given by

HZ = µB (gI Iz + gJ Jz) Bz (5)

for a magnetic field along the z direction. To determine the matrix elements in the coupled basis, we use the Wigner-Eckart theorem

F mF F 1 F ′ F, mF Jz F ′, mF′ = F J F ′ ( 1) − . (6) ⟨ | | ⟩ ⟨ || || ⟩ − mF 0 m′ ( − F )

Note that the 3j symbol ensures that only states with mF = mF′ are coupled. The reduced matrix element can be further reduced as

J +I+F +1 J ′ F ′ I F J F ′ = J J J ′ ( 1) ′ ′ (2F + 1)(2F + 1) . (7) ⟨ || || ⟩ ⟨ || || ⟩ − ′ FJ 1 { } √ The reduced matrix element in the above equation is easily shown to be

J J J ′ = δ J(J + 1)(2J + 1). (8) ⟨ || || ⟩ J,J′ √ Putting it all together:

2 41

F +F +J m +I+1 F, m J F ′, m′ = ( 1) ′ − F J(J + 1)(2J + 1) (2F + 1)(2F + 1) ⟨ F | z| F ⟩ − ′ F 1 F J F I √ √ ′ ′ ′ . (9) × mF 0 m′ FJ 1 ( − F ){ }

A similar equation holds for the Iz matrix element. [4]: def AngularElement(F,mF,Fp,mFp,J1,J2,alpha):

#To account for the CG coefficient phase, set alpha=0 for Jz and alpha=1␣ , for Iz →

if mF != mFp: return 0

if np.abs(F-Fp) > 1: return 0

if alpha == 0: #Matrix Element for Jz a = (-1)**(F+Fp+J1-mF+J2+1) elif alpha == 1: #Matrix Element of Iz a = (-1)**(2*F+J1-mF+J2+1)

b = np.sqrt(J1*(J1+1)*(2*Fp+1)*(2*F+1)*(2*J1+1)) c = wigner6j(J1,Fp,J2,F,J1,1)*wigner3j(F,1,Fp,-mF,0,mFp)

return a*b*c

def ZeemanHamiltonian(F,mF,J,I):

n = len(F) HZJ = np.zeros((n,n)) HZI = np.zeros((n,n))

for i in range(0,n): for j in range(0,i+1):

HZJ[i][j] = AngularElement(F[i],mF[i],F[j],mF[j],J,I,0) HZJ[j][i] = HZJ[i][j]

HZI[i][j] = AngularElement(F[i],mF[i],F[j],mF[j],I,J,1) HZI[j][i] = HZI[i][j]

return HZJ, HZI

3 42

1.0.2 Hamiltonian for 5S1/2 Ground State

[5]: F, mF = stateList([1,2])

J = 0.5 I = 1.5

muB = 0.001399624624 # Bohr magneton in MHz/G

### parameters for the ground state of Rb87 AhfsS12 = 3.41734130545215 # in GHz BhfsS12 = 0 gJS12 = 2.00233113 gI = -0.0009951414

### HhfsS12 = hyperfineHamiltonian(F,I,J,AhfsS12,BhfsS12) HZJS12, HZIS12 = ZeemanHamiltonian(F,mF,J,I) ###

def eigenValues5S12(B): HS12 = HhfsS12 + (gJS12*HZJS12+gI*HZIS12)*muB*B return np.linalg.eigvalsh(HS12)

BS12 = np.arange(0,15050,50) N = len(BS12)

EnergyS12 = np.zeros((N,len(F))) for k in range(N): EnergyS12[k,:] = eigenValues5S12(BS12[k])

[116]: ### Plot the results of the diagonalization

"""fig1 = plt.figure(figsize=(12,10)) ax10 = fig1.add_subplot(111) ax10.plot(BS12,EnergyS12)

ax10.set_xlabel("Magnetic Field (G)") ax10.set_ylabel("Energy (GHz)") ax10.set_xlim((0,600)) ax10.set_ylim((-7,5)) ax10.tick_params(axis='both', right=True, top=True, direction='in', length=10) #ax10.set_title(r"Zeeman shifts of the 5$^2$S$_{1/2}$ ground state") red_patch = mpatches.Patch(color='C0', label=r"$m_F = -1$") blue_patch = mpatches.Patch(color='C1', label=r"$m_F = 0$") c1 = mpatches.Patch(color='w', label=r"")

4 43

c2 = mpatches.Patch(color='C2', label=r"$m_F = 1$") c3 = mpatches.Patch(color='C3', label=r"$m_F = -2$") c4 = mpatches.Patch(color='C4', label=r"$m_F = -1$") c5 = mpatches.Patch(color='C5', label=r"$m_F = 0$") c6 = mpatches.Patch(color='C6', label=r"$m_F = 1$") c7 = mpatches.Patch(color='C7', label=r"$m_F = 2$") ax10.text(30, 3.3, r"$F = 2$", fontsize=24) ax10.text(30, -3.5, r"$F = 1$", fontsize=24) ax10.text(500, 1, r"$m_F = -2$", fontsize=18) ax10.text(500, 3.95, r"$m_F = +2$", fontsize=18) ax10.text(500, -3.4, r"$m_F = +1$", fontsize=18) ax10.text(500, -5.4, r"$m_F = -1$", fontsize=18) #ax10.legend(handles=[c7,c6,c5,c4,c3,c1, c2, blue_patch,␣ , red_patch],prop={'size': 15}, loc = 'center right', bbox_to_anchor=(0.45, 0. → , 45)) → plt.show()"""

print("Zero field energy levels: ", eigenValues5S12(0)) print("Covey field energy levels: ", eigenValues5S12(545.6))

('Zero field energy levels: ', array([-4.27167663, -4.27167663, -4.27167663, 2.56300598, 2.56300598, 2.56300598, 2.56300598, 2.56300598])) ('Covey field energy levels: ', array([-4.71219829, -4.35623504, -3.9599172 , 1.79962061, 2.2527664 , 2.64756439, 3.00200778, 3.32639135]))

1.0.3 Hamiltonian for 5P3/2 Ground State

[7]: F, mF = stateList([0,1,2,3])

J = 1.5 I = 1.5

muB = 0.001399624624 # Bohr magneton in MHz/G

### parameters for the 5P_{3/2} state of Rb87 AhfsP32 = 0.0847185 # in GHz BhfsP32 = 0.0124965 gJ32 = 1.3362 gI = -0.0009951414

HhfsP32 = hyperfineHamiltonian(F,I,J,AhfsP32,BhfsP32) HZJP32, HZIP32 = ZeemanHamiltonian(F,mF,J,I)

5 44

def eigenValues5P32(B): HP32 = HhfsP32 + (gJ32*HZJP32+gI*HZIP32)*muB*B return np.linalg.eigvalsh(HP32)

BP32 = np.arange(0,601,1) N = len(BP32)

EnergyP32 = np.zeros((N,len(F)))

for k in range(N): EnergyP32[k,:] = eigenValues5P32(BP32[k])

[117]: ### Plot the results of the diagonalization

"""fig2 = plt.figure(2,figsize=(12,10)) ax20 = fig2.add_subplot(111)

ax20.plot(BP32,EnergyP32,color='black')

ax20.set_xlabel("Magnetic Field (G)") ax20.set_ylabel("Energy (GHz)") ax20.set_xlim((0,600)) ax20.set_ylim((-2.1,2.1)) ax20.tick_params(axis='both', right=True, top=True, direction='in', length=10) #ax20.set_title(r"Zeeman shifts of the 5$^2$P$_{3/2}$ excited state") ax20.text(30, 0.5, r"$F = 3$", fontsize=24) ax20.text(30, -0.6, r"$F = 0$", fontsize=24)

ax20.text(500, 1.12, r"$m_J = +3/2$", fontsize=18) ax20.text(500, 0.3, r"$m_J = +1/2$", fontsize=18) ax20.text(500, -0.3, r"$m_J = -1/2$", fontsize=18) ax20.text(500, -1.1, r"$m_J = -3/2$", fontsize=18)

plt.show()"""

print("Excited Zero field energy levels: ", eigenValues5P32(0)) print("Excited Covey field energy levels: ", eigenValues5P32(545.6))

('Excited Zero field energy levels: ', array([-0.27669023, -0.21696473, -0.21696473, -0.21696473, -0.07252073, -0.07252073, -0.07252073, -0.07252073, -0.07252073, 0.20662777, 0.20662777, 0.20662777, 0.20662777, 0.20662777, 0.20662777, 0.20662777])) ('Excited Covey field energy levels: ', array([-1.71723288, -1.60991082,

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-1.48185471, -1.32278637, -0.5903014 , -0.53262197, -0.47699946, -0.42325711, 0.47463844, 0.51742376, 0.54260996, 0.55736414, 1.37288315, 1.49424787, 1.61596174, 1.7360419 ]))

Now we calculate the energy shift at the field B0. The Covey field is 545.6 G. Note that the 5S 5P bare transition frequency is 384.230 484 468 5 THz 1/2 → 3/2 [112]: B0 = 545.6

excitedStateShift = eigenValues5P32(B0)[-5] - eigenValues5P32(0.0)[-8] ### The␣ , numbers in brackets are the states that are shifting → groundStateShift = eigenValues5S12(B0)[0] - eigenValues5S12(0.0)[0] OffsetLock = excitedStateShift-groundStateShift

print(groundStateShift, excitedStateShift) print(OffsetLock + 0.080) #The added 0.08 GHz accounts for the AOM shift print(OffsetLock)

(-0.44052165361539775, 0.6298848730288757) 1.1504065266442736 1.0704065266442735

1.1 Transition Strengths & Linewidths

At any magnetic field, the optical transition strength is given by the square of matrix elements of the dipole moment operator. For an arbitrary magnetic field, the eigenstates are superpositions of F, m states. If ϕ and ϕ are the initial and final states, respectively, the matrix element is | F ⟩ | i⟩ | f ⟩

ϕ µ ϕ = ϕ F, m F, m µ F ′, m′ F ′, m′ ϕ , (10) ⟨ i| q| f ⟩ ⟨ i| F ⟩⟨ F | q| F ⟩⟨ F | f ⟩ F,m F ,m ∑F ∑′ F′

where we inserted the identity operator twice. We can now evaluate the matrix element in the F, mF basis using the methods from above (Wigner-Eckhart theorem and reduction of the reduced matrix| ⟩ element). The result is

ϕ µ ϕ = J µ J ′ ϕ F, m F ′, m′ ϕ (11) ⟨ i| q| f ⟩ ⟨ || || ⟩ ⟨ i| F ⟩⟨ F | f ⟩ F,m F ,m ∑F ∑′ F′ F m F 1 F ′ ( 1) − F (12) × − mF q m′ ( − F ) J F I ( 1)J+I+F ′+1 (2F + 1)(2F + 1) ′ ′ (13) × − ′ FJ 1 { } √

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The reduced matrix element J µ J ′ = 4.227(5)ea0 is a known quantity for the D2 transition of Rb.* But since we only care about⟨ || || relative⟩ strengths, we don’t need to worry about it. It scales all of the different transitions in the same way. The linewidth of a transition is related to the dipole moment and transition frequency via

3 1 ωif 2 γif = 3 ϕi µq ϕf (14) 3πϵ0 ~c0 |⟨ | | ⟩|

*The value of the reduced matrix element is taken from Steck, who uses a different convention for the Wigner-Eckhart theorem. To convert to the standard convention, we need to multiply the reduced matrix element by (2J ′ + 1)/(2J + 1) = √2 for the D2 transition. [10]: B0 = 545.6 √

FS12, mFS12 = stateList([1,2]) FP32, mFP32 = stateList([0,1,2,3])

### Generate the Hamiltonians, Eigenvalues, and Eigenenvectors HS12 = HhfsS12 + (gJS12*HZJS12+gI*HZIS12)*muB*B0 HP32 = HhfsP32 + (gJ32*HZJP32+gI*HZIP32)*muB*B0

ES12, VS12 = np.linalg.eigh(HS12) EP32, VP32 = np.linalg.eigh(HP32)

#Note that V[:,n] is the nth eigenvector

def transitionStrength(psiI, SI, psIF, SF, q): # This function calculates the matrix element for mixed states psiI and␣ , psiF (it does not return the square!!!) → # psi are a list of the mixing coefficients for the initial and final states # S should be a list of the form [F, m, J, I], where each is a list of all␣ , the states → # q = -1 for sigma+ transitions

FI = SI[0] mI = SI[1] JI = SI[2] II = SI[3]

FF = SF[0]

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mF = SF[1] JF = SF[2] IF = SF[3]

nI = len(psiI) nF = len(psiF)

X = 0 for k1 in range(nI): if psiI[k1] != 0: for k2 in range(nF): x = 0 x += psiI[k1]*psiF[k2] x *= (-1.0)**(FI[k1]-mI[k1])*wigner3j(FI[k1], 1, FF[k2],␣ , -mI[k1], q, mF[k2]) → x *= (-1.0)**(FF[k2]+JI+II+1)*np. , sqrt((2*FF[k2]+1)*(2*FI[k1]+1))*wigner6j(JF, FF[k2], IF, FI[k1], JI, 1) →

X += x

return X

[125]: nI = len(FS12) JS12 = 0.5 SI = [FS12, mFS12, JS12, I]

nF = len(FP32) JP32 = 1.5 SF = [FP32, mFP32, JP32, I]

transitionStrengths = np.zeros((3,nI, nF))

q = [-1,0,1]

for k in range(3): for i in range(nI): for f in range(i,nF): psiI = VS12[:,i] psiF = VP32[:,f]

transitionStrengths[k,i,f] = np.abs(transitionStrength(psiI, SI,␣ , psiF, SF, q[k]))**2 →

"""fig3 = plt.figure(figsize=(16,8),tight_layout=True) ax30 = fig3.add_subplot(131)

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ax31 = fig3.add_subplot(132) ax32 = fig3.add_subplot(133)

ax30.pcolormesh(transitionStrengths[0,:,:].T,cmap='YlOrRd', clim=(0,0.25)) ax31.pcolormesh(transitionStrengths[1,:,:].T,cmap='YlOrRd', clim=(0,0.25)) h3 = ax32.pcolormesh(transitionStrengths[2,:,:].T,cmap='YlOrRd', clim=(0,0.25))

ax30.set_xticks(np.arange(0.5,8.5,1)) ax30.set_xticklabels(np.arange(0,8,1)) ax30.set_yticks(np.arange(0.5,17.5,1)) ax30.set_yticklabels(np.arange(0,16,1)) ax30.set_ylim((0,16)) ax30.set_ylabel('Excited States') ax30.set_title(r'$\sigma^+$ transitions')

ax31.set_yticks([]) ax31.set_xticks(np.arange(0.5,8.5,1)) ax31.set_xticklabels(np.arange(0,8,1)) ax31.set_xlabel('Ground States') ax31.set_title(r'$\pi$ transitions')

ax32.set_yticks([]) ax32.set_xticks(np.arange(0.5,8.5,1)) ax32.set_xticklabels(np.arange(0,8,1)) ax32.set_title(r'$\sigma^-$ transitions')

#fig3.colorbar(h3)

plt.show()"""

[125]: "fig3 = plt.figure(figsize=(16,8),tight_layout=True)\nax30 = fig3.add_subplot(131)\nax31 = fig3.add_subplot(132)\nax32 = fig3.add_subplot(133 )\n\nax30.pcolormesh(transitionStrengths[0,:,:].T,cmap='YlOrRd', clim=(0,0.25))\nax31.pcolormesh(transitionStrengths[1,:,:].T,cmap='YlOrRd', clim=(0,0.25))\nh3 = ax32.pcolormesh(transitionStrengths[2,:,:].T,cmap='YlOrRd', clim=(0,0.25))\n\n\nax30.set_xticks(np.arange(0.5,8.5,1))\nax30.set_xticklabels( np.arange(0,8,1))\nax30.set_yticks(np.arange(0.5,17.5,1))\nax30.set_yticklabels( np.arange(0,16,1))\nax30.set_ylim((0,16))\nax30.set_ylabel('Excited States')\nax30.set_title(r'$\\sigma^+$ transitions')\n\n\nax31.set_yticks([])\na x31.set_xticks(np.arange(0.5,8.5,1))\nax31.set_xticklabels(np.arange(0,8,1))\nax 31.set_xlabel('Ground States')\nax31.set_title(r'$\\pi$ transitions')\n\nax32.se t_yticks([])\nax32.set_xticks(np.arange(0.5,8.5,1))\nax32.set_xticklabels(np.ara nge(0,8,1))\nax32.set_title(r'$\\sigma^-$ transitions')\n\n#fig3.colorbar(h3)\n\nplt.show()"

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[124]: ### Transitions as a function of frequency

initialStateG = 0 ## Initial ground state initialStateE = 11 ## Initial excited state

transitionsOutG = transitionStrengths[:,initialStateG,:] transitionsOutE = transitionStrengths[:,:,initialStateE]

energiesOutG = EP32 - ES12[initialStateG] energiesOutE = EP32[initialStateE] - ES12

"""fig4 = plt.figure(figsize=(15,20))

ax40 = fig4.add_subplot(211) ax40.plot(energiesOutG, transitionsOutG.T, 'o',markersize=12) ax40.set_xlabel("Transition Energy - 384.230 484 THz (GHz)") ax40.set_ylabel("Relative Transition Strength") ax40.set_title("Transitions out of Ground State {}, $B_0$ = {} G". , format(initialStateG, B0)) → ax40.legend(['$\sigma_+$','$\pi$','$\sigma_-$'])

ax41 = fig4.add_subplot(212) ax41.plot(energiesOutE, transitionsOutE.T, 'o',markersize=12) ax41.set_xlabel("Transition Energy - 384.230 484 THz (GHz)") ax41.set_ylabel("Relative Transition Strength") ax41.set_title("Transitions out of Excited State {}, $B_0$ = {} G". , format(initialStateE, B0)) → ax41.legend(['$\sigma_+$','$\pi$','$\sigma_-$'])

plt.show()"""

[124]: 'fig4 = plt.figure(figsize=(15,20))\n\nax40 = fig4.add_subplot(211)\nax40.plot(energiesOutG, transitionsOutG.T, \'o\',markersize=12)\nax40.set_xlabel("Transition Energy - 384.230 484 THz (GHz)")\nax40.set_ylabel("Relative Transition Strength")\nax40.set_title("Transitions out of Ground State {}, $B_0$ = {} G".format(initialStateG, B0))\nax40.legend([\'$\\sigma_+$\',\'$\\pi$\',\'$\\sigma_-$\'])\n\nax41 = fig4.add_subplot(212)\nax41.plot(energiesOutE, transitionsOutE.T, \'o\',markersize=12)\nax41.set_xlabel("Transition Energy - 384.230 484 THz (GHz)")\nax41.set_ylabel("Relative Transition Strength")\nax41.set_title("Transitions out of Excited State {}, $B_0$ = {} G".format(initialStateE, B0))\nax41.legend([\'$\\sigma_+$\',\'$\\pi$\',\'$\\sigma_-$\'])\n\nplt.show()'

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[123]: ### Linewidths as a function of frequency

nu0 = 384230.484 omegaG = 2*np.pi*(energiesOutG+nu0)*1E9 gammaG = 1.0/(3.0*np.pi*const.ep0)*omegaG**3.0/(const.hbar*const.c0**3. , 0)*transitionsOutG*(np.sqrt(2)*4.227*const.e*const.a0)**2.0 →

omegaE = 2*np.pi*(energiesOutE+nu0)*1E9 gammaE = 1.0/(3.0*np.pi*const.ep0)*omegaE**3.0/(const.hbar*const.c0**3. , 0)*transitionsOutE*(np.sqrt(2)*4.227*const.e*const.a0)**2.0 →

"""fig5 = plt.figure(figsize=(15,20))

ax50 = fig5.add_subplot(211) ax50.plot(energiesOutG, gammaG.T/(2*np.pi*1E6), 'o',markersize=12) ax50.set_xlabel("Transition Energy - 384.230 484 THz (GHz)") ax50.set_ylabel("$\gamma/2\pi$ (MHz)") ax50.set_title("Linewidths out of Ground State {}, $B_0$ = {} G". , format(initialStateG, B0)) → ax50.legend(['$\sigma_+$','$\pi$','$\sigma_-$'])

ax51 = fig5.add_subplot(212) ax51.plot(energiesOutE, gammaE.T/(2*np.pi*1E6), 'o',markersize=12) ax51.set_xlabel("Transition Energy - 384.230 484 THz (GHz)") ax51.set_ylabel("$\gamma/2\pi$ (MHz)") ax51.set_title("Linewidths out of Excited State {}, $B_0$ = {} G". , format(initialStateE, B0)) → ax51.legend(['$\sigma_+$','$\pi$','$\sigma_-$'])

plt.show()"""

[123]: 'fig5 = plt.figure(figsize=(15,20))\n\nax50 = fig5.add_subplot(211)\nax50.plot(energiesOutG, gammaG.T/(2*np.pi*1E6), \'o\',markersize=12)\nax50.set_xlabel("Transition Energy - 384.230 484 THz (GHz)")\nax50.set_ylabel("$\\gamma/2\\pi$ (MHz)")\nax50.set_title("Linewidths out of Ground State {}, $B_0$ = {} G".format(initialStateG, B0))\nax50.legend([\'$\\sigma_+$\',\'$\\pi$\',\'$\\sigma_-$\'])\n\nax51 = fig5.add_subplot(212)\nax51.plot(energiesOutE, gammaE.T/(2*np.pi*1E6), \'o\',markersize=12)\nax51.set_xlabel("Transition Energy - 384.230 484 THz (GHz)")\nax51.set_ylabel("$\\gamma/2\\pi$ (MHz)")\nax51.set_title("Linewidths out of Excited State {}, $B_0$ = {} G".format(initialStateE, B0))\nax51.legend([\'$\\sigma_+$\',\'$\\pi$\',\'$\\sigma_-$\'])\n\nplt.show()'

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